Euler's Criterion of prime order in PID case

TL;DR

This paper extends Euler's criterion to prime order l in the context of cyclotomic integer rings that are principal ideal domains, using Jacobi sums to establish necessary conditions.

Contribution

It introduces a new Euler's criterion for prime order l in PID cases of cyclotomic integer rings based on Jacobi sums.

Findings

Established Euler's criterion for prime order l in PID cyclotomic rings.

Derived conditions involving Jacobi sums for the criterion.

Enhanced understanding of cyclotomic extensions in algebraic number theory.

Abstract

In this paper we establish the Eulers criterion of order l (a prime) when the ring of integers in the cyclotomic extension of Q of order l is a PID. Conditions are obtained in terms of Jacobi sums of order l.